आव्यूह

vThe essence of mathematics lies in its freedom — CANTOR _v

3.1

Hkwfedk

(Introduction)

xf.kr dh fofo/ 'kk[kkvksa esa vkO;wg osQ Kku dh vko';drk iM+rh gSA vkO;wg] xf.kr osQ lokZf/d 'kfDr'kkyh lk/uksa esa ls ,d gSA vU; lh/h&lknh fof/;ksa dh rqyuk esa ;g xf.krh; lk/u gekjs dk;Z dks dkiQh gn rd ljy dj nsrk gSA jSf[kd lehdj.kksa osQ fudk; dks gy djus osQ fy, laf{kIr rFkk ljy fof/;k¡ izkIr djus osQ iz;kl osQ ifj.kkeLo:i vkO;wg dh ladYiuk dk fodkl gqvkA vkO;wgksa dks osQoy jSf[kd lehdj.kksa osQ fudk; osQ xq.kkadksa dks izdV djus osQ fy, gh ugha iz;ksx fd;k tkrk gS] vfirq vkO;wgksa dh mi;ksfxrk bl iz;ksx ls dgha vf/d gSA vkO;wg laosQru rFkk lafØ;kvksa dk iz;ksx O;fDrxr oaQI;wVj osQ fy, bysDVªkfud LizsM'khV izksxzkeksa

(Electronic Spreadsheet Programmes) esa fd;k tkrk gS] ftldk iz;ksx] Øe'k% okf.kT; rFkk

foKku osQ fofHkUu {ks=kksa esa gksrk gS] tSls] ctV (Budgeting)] foØ; cfgosZ'ku (Sales Projection)] ykxr vkdyu (Cost Estimation)] fdlh iz;ksx osQ ifj.kkeksa dk fo'ys"k.k bR;kfnA

blosQ vfrfjDr vusd HkkSfrd lafØ;k,¡ tSls vko/Zu (Magnification)] ?kw.kZu (Rotation) rFkk

fdlh lery }kjk ijkorZu (Reflection) dks vkO;wgksa }kjk xf.krh; <ax ls fu:fir fd;k tk ldrk

gSA vkO;wgksa dk iz;ksx xw<+ysf[kdh (Cryptography) esa Hkh gksrk gSA bl xf.krh;

lk/u dk iz;ksx u osQoy foKku dh gh oqQN 'kk[kkvksa rd lhfer gS] vfirq bldk iz;ksx vuqoaf'kdh] vFkZ'kkL=k] vk/qfud euksfoKku rFkk vkS|kSfxd izca/u esa Hkh fd;k tkrk gSA

bl vè;k; esa vkO;wg rFkk vkO;wg chtxf.kr (Matrix algebra) osQ vk/kjHkwr fl¼karksa ls

voxr gksuk] gesa #fpdj yxsxkA

3.2

vkO;wg

(Matrix)

eku yhft, fd ge ;g lwpuk O;Dr djuk pkgrs gSa fd jk/k osQ ikl 15 iqfLrdk,¡ gSaA bls ge

[15] :i esa] bl le> osQ lkFk O;Dr dj ldrs gSa] fd [ ] osQ vanj fyf[kr la[;k jk/k osQ ikl

iqfLrdkvksa dh la[;k gSA vc ;fn gesa ;g O;Dr djuk gS fd jk/k osQ ikl 15 iqfLrdk,¡ rFkk 6 dyesa gSa] rks bls ge [15 6] izdkj ls] bl le> osQ lkFk O;Dr dj ldrs gSa fd [ ] osQ vanj

dh izFke izfof"V jk/k osQ ikl dh iqfLrdkvksa dh la[;k] tcfd f}rh; izfof"V jk/k osQ ikl dyeksa

vè;k;

3

dh la[;k n'kkZrh gSA vc eku yhft, fd ge jk/k rFkk mlosQ nks fe=kksa iQksSft;k rFkk fleju osQ ikl dh iqfLrdkvksa rFkk dyeksa dh fuEufyf[kr lwpuk dks O;Dr djuk pkgrs gSa%

jk/k osQ ikl 15 iqfLrdk,¡ rFkk 6 dye gSa]

iQkSft;k osQ ikl 10 iqfLrdk,¡ rFkk 2 dye gSa]

fleju osQ ikl 13 iqfLrdk,¡ rFkk 5 dye gSa]

vc bls ge lkjf.kd :i esa fuEufyf[kr izdkj ls O;ofLFkr dj ldrs gSa%

iqfLrdk dye

jk/k 15 6

iQkSft;k 10 2

fleju 13 5

bls fuEufyf[kr <ax ls O;Dr dj ldrs gSa%

vFkok

jk/k iQkSft;k fleju

iqfLrdk 15 10 13

dye 6 2 5

ftls fuEufyf[kr <ax ls O;Dr dj ldrs gSa%

fleju osQ ikl dyeksa dh la[;k izdV djrh gSaA blh izdkj] nwljh izdkj dh O;oLFkk esa izFke iaafDr dh izfof"V;k¡ Øe'k% jk/k] iQkSft;k rFkk fleju osQ ikl iqfLrdkvksa dh la[;k izdV djrh gSaA f}rh; iafDr dh izfof"V;k¡ Øe'k% jk/k] iQkSft;k rFkk fleju osQ ikl dyeksa dh la[;k izdV djrh gSaA mi;qZDr izdkj dh O;oLFkk ;k izn'kZu dks vkO;wg dgrs gSaA vkSipkfjd :i ls ge vkO;wg dks fuEufyf[kr izdkj ls ifjHkkf"kr djrs gSa%

ifjHkk"kk 1 vkO;wg la[;kvksa ;k iQyuksa dk ,d vk;rkdkj Øe&foU;kl gSA bu la[;kvksa ;k iQyuksa

dks vkO;wg osQ vo;o vFkok izfof"V;k¡ dgrs gSaA

vkO;wg dks ge vaxzsth o.kZekyk osQ cM+s (Capital) v{kjksa }kjk O;Dr djrs gSaA vkO;wgksa osQ oqQN

mnkgj.k fuEufyf[kr gSa%

5 – 2

A 0 5

3 6

 

 

=  

 

 

,

1

2 3

2

B 3.5 –1 2

5

3 5

7 i

 _{+ −} 

 

 

=  

 

 

 

,

3

1 3

C

cos sin 2 tan

x x

x x x

 + 

=  

+

 

mi;qZDr mnkgj.kksa esa {kSfrt js[kk,¡ vkO;wg dh iafDr;k¡ (Rows) vksj ÅèoZ js[kk,¡ vkO;wg osQ

LraHk (Columns) dgykrs gSaA bl izdkj A esa 3 iafDr;k¡ rFkk 2 LraHk gSa vkSj B esa 3 iafDr;k¡ rFkk

3 LraHk tcfd C esa 2 iafDr;k¡ rFkk 3 LraHk gSaA

3.2.1 vkO;wg dh dksfV (Order of a matrix)

miafDr;ksa rFkk nLraHkksa okys fdlh vkO;wg dks m × ndksfV (order) dk vkO;wg vFkok osQoy

m × nvkO;wg dgrs gSa A vr,o vkO;wgksa osQ mi;ZqDr mnkgj.kksa osQ lanHkZ esa A, ,d 3 × 2 vkO;wg]

B ,d 3 × 3 vkO;wg rFkk C, ,d 2 × 3 vkO;wg gSaA ge ns[krs gSa fd A esa 3 × 2 = 6 vo;o gS

vkSj B rFkk C esa Øe'k% 9 rFkk 6 vo;o gSaA

lkekU;r%] fdlh m × nvkO;wg dk fuEufyf[kr vk;krkdkj Øe&foU;kl gksrk gS%

bl izdkj ioha iafDr osQ vo;o a_i1, a_i2, a_i3,..., a_ingSa] tcfd josa LraHk osQ vo;oa_1j, a_2j,

a_3j,..., a_mj gSaA

lkekU;r% a_ij, ioha iafDr vkSj josa LraHk esa vkus okyk vo;o gksrk gSA ge bls A dk (i, j)ok¡

vo;o Hkh dg ldrs gSaA fdlh m × nvkO;wg esa vo;oksa dh la[;k mngksrh gSA

A

fVIi.kh bl vè;k; esa]

1. ge fdlh m × n dksfV osQ vkO;wg dks izdV djus osQ fy,] laosQr A = [a_ij]_{m × n}dk iz;ksx

djsaxsA

2. ge osQoy ,sls vkO;wgksa ij fopkj djsaxs] ftuosQ vo;o okLrfod la[;k,¡ gSa vFkok

okLrfod ekuksa dks xzg.k djus okys iQyu gSaA

ge ,d lery osQ fdlh fcanq (x, y) dks ,d vkO;wg (LraHk vFkok iafDr) }kjk izdV dj

ldrs gSa] tSls x

y

   

  (vFkok [x, y])ls] mnkgj.kkFkZ] fcanq P(0, 1), vkO;wg fu:i.k esa 0 P

1   =  

  ;k

[0 1] }kjk izdV fd;k tk ldrk gSA

è;ku nhft, fd bl izdkj ge fdlh can jSf[kd vko`Qfr osQ 'kh"kks± dks ,d vkO;wg osQ :i esa fy[k ldrs gSaA mnkgj.k osQ fy, ,d prqHkZt ABCD ij fopkj dhft,] ftlosQ 'kh"kZ Øe'k% A (1, 0), B (3, 2), C (1, 3), rFkk D (–1, 2) gSaA

vc] prqHkZqt ABCD vkO;wg :i esa fuEufyf[kr izdkj ls fu:fir fd;k tk ldrk gS%

2 4

A B C D 1 3 1 1 X

0 2 3 2 _× −

 

=  

  ;k

4 2

A 1 0 B 3 2 Y

C 1 3

D 1 2 _×

 

 

 

=

 

₋ 

 

vr% vkO;wgksa dk iz;ksx fdlh lery esa fLFkr T;kferh; vko`Qfr;ksa osQ 'kh"kks± dks fu:fir djus osQ fy, fd;k tk ldrk gSA

vkb, vc ge oqQN mnkgj.kksa ij fopkj djsaA

mnkgj.k 1rhu iSQfDVª;ksa I, II rFkk III esa iq#"k rFkk efgyk dfeZ;ksa ls lacaf/r fuEufyf[kr lwpuk

iq#"k dehZ efgyk dehZ

I 30 25

II 25 31

III 27 26

mi;qZDr lwpuk dks ,d 3 × 2 vkO;wg esa fu:fir dhft,A rhljh iafDr vkSj nwljs LraHk okyh

izfof"V D;k izdV djrh gS\

gy iznÙk lwpuk dks 3 × 2 vkO;wg osQ :i esa fuEufyf[kr izdkj ls fu:fir fd;k tk ldrk gS%

30 25

A 25 31

27 26

 

 

=  

 

 

rhljh iafDr vkSj nwljs LraHk dh izfof"V iSQDVªh&III dkj[kkus esa efgyk dk;ZdrkZvksa dh la[;k

izdV djrh gSA

mnkgj.k 2 ;fn fdlh vkO;wg esa 8 vo;o gSa] rks bldh laHko dksfV;k¡ D;k gks ldrh gSa\

gy gesa Kkr gS fd] ;fn fdlh vkO;wg dh dksfV m × ngS rks blesa mnvo;o gksrs gSaA vr,o

8 vo;oksa okys fdlh vkO;wg osQ lHkh laHko dksfV;k¡ Kkr djus osQ fy, ge izko`Qr la[;kvksa osQ mu lHkh Øfer ;qXeksa dks Kkr djsaxs ftudk xq.kuiQy 8 gSA

vr% lHkh laHko Øfer ;qXe (1, 8), (8, 1), (4, 2), (2, 4) gSaA

vr,o laHko dksfV;k¡ 1 × 8, 8 ×1, 4 × 2, 2 × 4 gSaA

mnkgj.k 3 ,d ,sls 3 × 2 vkO;wg dh jpuk dhft,] ftlosQ vo;o 1| 3 | 2

ij

a = i− j }kjk

iznÙk gSaA

gy ,d 3 × 2 vkO;wg] lkekU;r% bl izdkj gksrk gS%

11 12 21 22 31 32

A

a a

a a

a a

 

 

=  

 

 

vc] 1| 3 |

2

ij

a = i− j , i = 1, 2, 3rFkk j = 1, 2

blfy,

11

1

|1 3.1| 1 2

a = − = ₁₂ 1|1 3.2 | 5

2 2

21

1 1

| 2 3.1|

2 2

a = − = ₂₂ 1| 2 3.2 | 2

2

a = − =

31

1

| 3 3.1| 0 2

a = − = ₃₂ 1| 3 3.2 | 3

2 2

a = − =

vr% vHkh"V vkO;wg

5 1

2 1

A 2

2 3 0

2

 

 

 

=  

 

 

 

gSA

3.3

vkO;wgksa osQ izdkj

(Types of Matrices)

bl vuqPNsn esa ge fofHkUu izdkj osQ vkO;wgksa dh ifjppkZ djsaxsA

(i) LraHk vkO;wg (Column matrix)

,d vkO;wg] LraHk vkO;wg dgykrk gS] ;fn mlesa osQoy ,d LraHk gksrk gSA mnkgj.k osQ

fy,

0

3

A 1

1/2

 

 

 

 

= −

 

 

 

, 4 × 1 dksfV dk ,d LraHk vkO;wg gSA O;kid :i ls] A= [a_ij]_{m × 1} ,d

m × 1 dksfV dk LraHk vkO;wg gSA

(ii) iafDr vkO;wg (Row matrix)

,d vkO;wg] iafDr vkO;wg dgykrk gS] ;fn mlesa osQoy ,d iafDr gksrh gSA

mnkgj.k osQ fy,

1 4

1

B 5 2 3

2 ×

 

= −_{ }

  , 1×4 dksfV dk ,d iafDr vkO;wg gSAO;kid

:i ls] B = [b_ij]_{1 × n},d 1 × n dksfV dk iafDr vkO;wg gSA

(iii) oxZ vkO;wg (Square matrix)

,d vkO;wg ftlesa iafDr;ksa dh la[;k LraHkksa dh la[;k osQ leku gksrh gS] ,d oxZ vkO;wg

‘n’ dk oxZ vkO;wg dgrs gSaA mnkgj.k osQ fy,

3 1 0

3

A 3 2 1

2

4 3 1

−

 

 

 

=

 

 ₋ 

 

,d 3 dksfV dk oxZ

vkO;wg gSA O;kid :i ls A = [a_ij]_{m × m},d m dksfV dk oxZ vkO;wg gSA

A

fVIi.kh ;fn A = [a_ij] ,d n dksfV dk oxZ vkO;wg gS] rks vo;oksa (izfof"V;k¡) a₁₁, a₂₂, ..., a_nndks vkO;wg A osQ fod.kZ osQ vo;o dgrs gSaA

vr% ;fn

1 3 1

A 2 4 1

3 5 6

−

 

 

=_ − _

 

 

gS rks A osQ fod.kZ osQ vo;o 1] 4] 6 gSaA

(iv) fod.kZ vkO;wg (Diagonal matrix)

,d oxZ vkO;wg B = [b_ij]_{m × m}fod.kZ vkO;wg dgykrk gS] ;fn fod.kZ osQ vfrfjDr blosQ

vU; lHkh vo;o 'kwU; gksrs gSa vFkkZr~] ,d vkO;wg B = [b_ij]_{m × m}fod.kZ vkO;wg dgykrk

gS] ;fn b_ij = 0, tc i≠jgksA

mnkgj.kkFkZ A = [4], B 1 0

0 2 −

 

=  

 ,

1.1 0 0

C 0 2 0

0 0 3

−

 

 

=  

 

 

, Øe'k% dksfV 1] 2 rFkk 3 osQ

fod.kZ vkO;wg gSaA

(v) vfn'k vkO;wg (Scalar matrix)

,d fod.kZ vkO;wg] vfn'k vkO;wg dgykrk gS] ;fn blosQ fod.kZ osQ vo;o leku gksrs

gSa] vFkkZr~] ,d oxZ vkO;wg B = [b_ij]_{n × n}vfn'k vkO;wg dgykrk gS] ;fn

b_ij = 0, tc i ≠j

b_ij = k, tc i = j, tgk¡ kdksbZ vpj gSA

mnkgj.kkFkZ]

A = [3], B 1 0

0 1

−

 

=  ₋ 

 ,

3 0 0

C 0 3 0

0 0 3

 

 

=  

 

 

Øe'k%

(vi) rRled vkO;wg (Identity matrix)

,d oxZ vkO;wg] ftlosQ fod.kZ osQ lHkh vo;o 1 gksrs gSa rFkk 'ks"k vU; lHkh vo;o 'kwU; gksrs gSa] rRled vkO;wg dgykrk gSA nwljs 'kCnksa esa] oxZ vkO;wg A = [a_ij]_{n × n},d rRled

vkO;wg gS] ;fn a i j

i j

ij =

= ≠ 

 

1 0

;fn ;fn

ge] n dksfV osQ rRled vkO;wg dks I_n }kjk fu:fir djrs gSaA tc lanHkZ ls dksfV Li"V gksrh

gS] rc bls ge osQoy I ls izdV djrs gSaA

mnkgj.k osQ fy, [1], 1 0

0 1

 

 

 ,

1 0 0

0 1 0 0 0 1

 

 

 

 

 

Øe'k% dksfV 1, 2 rFkk 3 osQ rRled vkO;wg gSaA

è;ku nhft, fd ;fn k = 1 gks rks, ,d vfn'k vkO;wg] rRled vkO;wg gksrk gS] ijarq izR;sd

rRled vkO;wg Li"Vr;k ,d vfn'k vkO;wg gksrk gSA

(vii) 'kwU; vkO;wg (Zero matrix)

,d vkO;wg] 'kwU; vkO;wg vFkok fjDr vkO;wg dgykrk gS] ;fn blosQ lHkh vo;o 'kwU;

gksrs gSaA

mnkgj.kkFkZ] [0],

0 0 0 0

 

 



, 0 0 0

0 0 0

 

 



, [0, 0] lHkh 'kwU; vkO;wg gSaA ge 'kwU; vkO;wg dks

O }kjk fu:fir djrs gSaA budh dksfV;k¡] lanHkZ }kjk Li"V gksrh gSaA

3.3.1 vkO;wgksa dh lekurk (Equality of matrices)

ifjHkk"kk 2nks vkO;wg A = [a_ij] rFkk B = [b_ij] leku dgykrs gSa] ;fn

(i) os leku dksfV;ksa osQ gksrs gksa] rFkk

(ii) A dk izR;sd vo;o] B osQ laxr vo;o osQ leku gks] vFkkZr~ irFkk josQ lHkh ekuksa osQ

fy, a_ij = b_ijgksa

mnkgj.k osQ fy,] 2₀ 3₁ 2₀ 3₁



 



 



 

 

rFkk leku vkO;wg gSa ¯drq 3₀ 2₁ 2₀ 3₁



 



 



 

  rFkk leku

;fn

1.5 0

2 6

3 2

x y

z a

b c

−

 

 

 

 _{ =}

 

 

 

 

   

, rks x = – 1.5, y = 0, z = 2, a = 6, b = 3, c = 2

mnkgj.k 4;fn

3 4 2 7 0 6 3 2

6 1 0 6 3 2 2

3 21 0 2 4 21 0

x z y y

a c

b b

+ + − −

   

 _{− −}  _{= − − +} 

   

 − −   + − 

   

gks rks a, b, c, x, yrFkk zosQ eku Kkr dhft,A

gy pw¡fd iznÙk vkO;wg leku gSa] blfy, buosQ laxr vo;o Hkh leku gksaxsA laxr vo;oksa dh

rqyuk djus ij gesa fuEufyf[kr ifj.kke izkIr gksrk gS%

x + 3 = 0, z + 4 = 6, 2y – 7 = 3y – 2

a – 1 = – 3, 0 =2c + 2 b – 3 = 2b + 4,

bUgsa ljy djus ij gesa izkIr gksrk gS fd

a = – 2, b = – 7, c = – 1, x = – 3, y = –5, z = 2

mnkgj.k 5 ;fn

2 2 4 3

5 4 3 11 24

a b a b

c d c d

+ − −

   

=

 _{− +}   

   gks rks a, b, c, rFkk d osQ eku Kkr dhft,A

gy nks vkO;wgksa dh lekurk dh ifjHkk"kk }kjk] laxr vo;oksa dks leku j[kus ij gesa izkIr gksrk gS fd

2a + b = 4 5c – d = 11

a – 2b = – 3 4c + 3d = 24

bu lehdj.kksa dks ljy djus ij a = 1, b = 2, c = 3 rFkkd = 4 izkIr gksrk gSA

iz'ukoyh 3-1

1. vkO;wg

2 5 19 7

5

A 35 2 12

2 17

3 1 5

 − 

 

 

= −

 

 ₋ 

 

, osQ fy, Kkr dhft,%

(i) vkO;wg dh dksfV (ii) vo;oksa dh la[;k

2. ;fn fdlh vkO;wg esa 24 vo;o gSa rks bldh laHko dksfV;k¡ D;k gSa\ ;fn blesa 13 vo;o

gksa rks dksfV;k¡ D;k gksaxh\

3. ;fn fdlh vkO;wg esa 18 vo;o gSa rks bldh laHko dksfV;k¡ D;k gSa\ ;fn blesa 5 vo;o

gksa rks D;k gksxk\

4. ,d 2 × 2 vkO;wg A = [a_ij] dh jpuk dhft, ftlosQ vo;o fuEufyf[kr izdkj ls iznÙk gSa

(i)

2

( )

2

ij

i j

a = + (ii) ij

i a

j

= (iii)

2

( 2 ) 2

ij

i j

a = +

5. ,d 3 × 4 vkO;wg dh jpuk dhft, ftlosQ vo;o fuEufyf[kr izdkj ls izkIr gksrs gSa%

(i) 1| 3 |

2

ij

a = − +i j (ii) a_ij =2i− j

6. fuEufyf[kr lehdj.kksa ls x, yrFkk zosQ eku Kkr dhft,%

(i) 4 3

5 1 5

y z x

   

=

   

    (ii)

2 6 2

5 5 8

x y z xy

+

  ₌ 

 ₊   

   (iii)

9 5

7 x y z

x z

y z

+ +

   

 ₊   ₌

   

 +   

   

7. lehdj.k 2 1 5

2 3 0 13

a b a c

a b c d

− + −

  ₌ 

 _{− +}   

    ls a, b, crFkk dosQ eku Kkr dhft,A

8. A = [a_ij]_{m × n\},d oxZ vkO;wg gS ;fn

(A) m < n (B) m > n (C) m = n (D) buesa ls dksbZ ugha

9. xrFkkyosQ iznÙk fdu ekuksa osQ fy, vkO;wgksa osQ fuEufyf[kr ;qXe leku gSa\

3 7 5

1 2 3 x

y x

+

 

 _{+ −} 

  ,

0 2

8 4

y−

 

 

 

(A) 1 , 7

3

x=− y= (B) Kkr djuk laHko ugha gS

(C) y = 7 , 2 3

x=− (D) 1 , 2

3 x

3 y

− −

= = .

10. 3 × 3 dksfV osQ ,sls vkO;wgksa dh oqQy fdruh la[;k gksxh ftudh izR;sd izfof"V 0 ;k 1 gS?

(A) 27 (B) 18 (C) 81 (D) 512

3.4

vkO;wgksa ij lafØ;k,¡

(Operations on Matrices)

3.4.1 vkO;wgksa dk ;ksx (Addition of matrices)

eku yhft, fd iQkfrek dh LFkku A rFkk LFkku B ij nks iSQfDVª;k¡ gSaA izR;sd iSQDVªh esa yM+dksa

rFkk yM+fd;ksa osQ fy,] [ksy osQ twrs] rhu fHkUu&fHkUu ewY; oxks±] Øe'k% 1] 2 rFkk 3 osQ curs gSaA izR;sd iSQDVªh esa cuus okys twrksa dh la[;k uhps fn, vkO;wgksa }kjk fu:fir gSa%

eku yhft, fd iQkfrek izR;sd ewY; oxZ esa cuus okys [ksy osQ twrksa dh oqQy la[;k tkuuk pkgrh gSaA vc oqQy mRiknu bl izdkj gS%

ewY; oxZ 1 : yM+dksa osQ fy, (80 + 90), yM+fd;ksa osQ fy, (60 + 50)

ewY; oxZ 2 : yM+dksa osQ fy, (75 + 70), yM+fd;ksa osQ fy, (65 + 55)

ewY; oxZ 3 : yM+dksa osQ fy, (90 + 75), yM+fd;ksa osQ fy, (85 + 75)

vkO;wg osQ :i esa bls bl izdkj izdV dj ldrs gSa

80 90 60 50

75 70 65 55

90 75 85 75

+ +

 

 _{+ +} 

 

 + + 

 

;g u;k vkO;wg] mi;qZDr nks vkO;wgksa dk ;ksxiQy gSA ge ns[krs gSa fd nks vkO;wgksa dk ;ksxiQy,

iznÙk vkO;wgksa osQ laxr vo;oksa dks tksM+us ls izkIr gksus okyk vkO;wg gksrk gSA blosQ vfrfjDr] ;ksx osQ fy, nksuksa vkO;wgksa dks leku dksfV dk gksuk pkfg,A

bl izdkj] ;fn 11 12 13

21 22 23

A a a a

a a a

 

=  

  ,d 2 × 3 vkO;wg gS rFkk

11 12 13 21 22 23

B b b b

b b b

 

=  

  ,d

vU; 2 × 3 vkO;wg gS] rks ge 11 11 12 12 13 13 21 21 22 22 23 23

A + B a b a b a b

a b a b a b

+ + +

 

=  _{+ + +} 

 }kjk ifjHkkf"kr djrs gSaA

O;kid :i ls] eku yhft, fd A = [a_ij] rFkk B = [b_ij] nks leku dksfV] m × nokys vkO;wg

gSa rks A rFkk B nksuksa vkO;wgksa dk ;ksxiQy] vkO;wg C = [c_ij]_{m × n}, }kjk ifjHkkf"kr gksrk gS] tgk¡

mnkgj.k 6 A 3 1 1

2 3 0

 ₋ 

=  

  rFkk

2 5 1

B ₁

2 3 2

 

 

= ₋ 

 

 

gS rks A + B Kkr dhft,A

gy D;ksafd A rFkk B leku dksfV 2 × 3 okys vkO;wg gSa] blfy, A rFkk B dk ;ksx ifjHkkf"kr

gS] vkSj

2 3 1 5 1 1 2 3 1 5 0

A + B _{1 1}

2 2 3 3 0 0 6

2 2

 _{+ + −}   _{+ +} 

   

=_{  }= _

− + +

   

   

}kjk izkIr gksrk gSA

A

fVIi.kh

1. ge bl ckr ij cy nsrs gSa fd ;fn A rFkk B leku dksfV okys vkO;wg ugha gSa rks

A + B ifjHkkf"kr ugha gSA mnkgj.kkFkZ A 2 3 1 0

 

=  

 ,

1 2 3

B ,

1 0 1

 

=  

  rks A + B ifjHkkf"kr

ugha gSA

2. ge ns[krs gSa fd vkO;wgksa dk ;ksx] leku dksfV okys vkO;wgksa osQ leqPp; esa f}vk/kjh

lafØ;k dk ,d mnkgj.k gSA

3.4.2 ,d vkO;wg dk ,d vfn'k ls xq.ku (Multiplication of a matrix by a scalar)

vc eku yhft, fd i+Qkfrek us A ij fLFkr iSQDVªh esa lHkh ewY; oxZ osQ mRiknu dks nks xquk dj

fn;k gS (lanHkZ 3-4-1)

A ij fLFkr iSQDVªh esa mRiknu dh la[;k uhps fn, vkO;wg esa fn[kykbZ xbZ gSA

.

2 80 2 60 1

2 2 75 2 65

3 2 90 2 85

× ×

 

 _{× ×} 

 

 × × 

 

yM+osQ yM+fd;k¡

bls vkO;wg :i esa ,

160 120

150 130

180 170

 

 

 

 

 

izdkj ls fu:fir dj ldrs gSaA ge ns[krs gSa fd ;g

u;k vkO;wg igys vkO;wg osQ izR;sd vo;o dks 2 ls xq.kk djus ij izkIr gksrk gSA

O;kid :i esa ge] fdlh vkO;wg osQ ,d vfn'k ls xq.ku dks] fuEufyf[kr izdkj ls ifjHkkf"kr djrs gSaA ;fn A = [a_ij]_{m × n},d vkO;wg gS rFkk k ,d vfn'k gS rks kA ,d ,slk vkO;wg gS ftls A osQ izR;sd vo;o dks vfn'k kls xq.kk djosQ izkIr fd;k tkrk gSA

nwljs 'kCnksa esa] kA = k[a_ij]_{m × n} = [k(a_ij)]_{m × n}, vFkkZr~ kA dk (i, j)ok¡ vo;o] irFkk j

osQ gj laHko eku osQ fy,] ka_ijgksrk gSA

mnkgj.k osQ fy,] ;fn A =

3 1 1.5

5 7 3

2 0 5

 

 

−

 

 

 

gSrks

3A =

3 1 1.5 9 3 4.5

3 5 7 3 3 5 21 9

2 0 5 6 0 15

   

   

− = −

   

   

   

vkO;wg dk ½.k vkO;wg (Negative of a matrix)fdlh vkO;wg A dk ½.k vkO;wg –A

ls fu:fir gksrk gSA ge –A dks – A = (– 1) A }kjk ifjHkkf"kr djrs gSaA

mnkgj.kkFkZ] eku yhft, fd A = 3 1

5 x

 

₋ 

 , rks – A fuEufyf[kr izdkj ls izkIr gksrk gS

– A = (– 1)A ( 1) 3 1 3 1

5 x 5 x

− −

   

= − _{  }= _

− −

   

vkO;wgksa dk varj (Difference of matrices);fn A = [a_ij], rFkk B = [b_ij] leku dksfV

ekuksa osQ fy, d_ij= a_ij– b_ij gS, }kjk ifjHkkf"kr gksrk gSA nwljs 'kCnksa esa] D = A – B = A + (–1) B,

vFkkZr~ vkO;wg A rFkk vkO;wg – B dk ;ksxiQyA

mnkgj.k 7 ;fn A 1 2 3 B 3 1 3

2 3 1 1 0 2

−

   

=_{ } =_{ }

−

 rFkk  gSarks 2A – B Kkr dhft,A

gy ge ikrs gSa

2A – B = 2 1 2 3 3 1 3

2 3 1 1 0 2

−

   

−

  ₋ 

   

=

2 4 6 3 1 3

4 6 2 1 0 2

− −

   

+

   ₋ 

   

= 2 3 4 1 6 3 1 5 3

4 1 6 0 2 2 5 6 0

− + − −

   

=

 _{+ + −}   

   

3.4.3 vkO;wgksa osQ ;ksx osQ xq.k/eZ (Properties of matrix addition)

vkO;wgksa osQ ;ksx dh lafØ;k fuEufyf[kr xq.k/eks± (fu;eksa) dks larq"V djrh gS%

(i) Øe&fofues; fu;e (Commutative Law) ;fn A = [a_ij], B = [b_ij] leku dksfV

m × n, okys vkO;wg gSa] rks A + B = B + A gksxkA

vc A + B = [a_ij] + [b_ij] = [a_ij + b_ij]

= [b_ij+ a_ij] (la[;kvksa dk ;ksx Øe&fofues; gSA)

= ([b_ij] + [a_ij]) = B + A

(ii) lkgp;Z fu;e (Associative Law) leku dksfV m × n okys fdUgha Hkh rhu vkO;wgksa

A = [a_ij], B = [b_ij], C = [c_ij] osQ fy, (A + B) + C = A + (B + C)

vc (A + B) + C = ([a_ij] + [b_ij]) + [c_ij]

= [a_ij + b_ij] + [c_ij] = [(a_ij + b_ij) + c_ij]

= [a_ij + (b_ij + c_ij)] (D;ksa ?)

= [a_ij] + [(b_ij+ c_ij)] = [a_ij] + ([b_ij] + [c_ij]) = A + (B + C)

(iii) ;ksx osQ rRled dk vfLrRo (Existence of additive identity)eku yhft, fd

A = [a_ij] ,d m × nvkO;wg gS vkSj O ,d m × n'kwU; vkO;wg gS] rks A+O = O+A= A

gksrk gSAnwljs 'kCnksa esa] vkO;wgksa osQ ;ksx lafØ;k dk rRled 'kwU; vkO;wg O gSA

(iv) ;ksx osQ izfrykse dk vfLrRo (The existence of additive inverse) eku yhft,

fd A + (– A) = (– A) + A= O]vr,o vkO;wg – A, vkO;wg A dk ;ksx osQ varxZr

izfrykse vkO;wg vFkok ½.k vkO;wg gSA

3.4.4 ,d vkO;wg osQ vfn'k xq.ku osQ xq.k/eZ (Properties of scalar multiplication of a matrix)

;fn A = [a_ij] rFkk B = [b_ij] leku dksfV m × n, okys nks vkO;wg gSa vkSj krFkk lvfn'k gSa] rks

(i) k(A +B) = k A + kB, (ii) (k + l)A = k A + l A

vc] A = [a_ij]_{m × n}, B = [b_ij]_{m × n}, vkSj krFkk l vfn'k gSa] rks

(i) k (A + B) = k ([a_ij] + [b_ij])

= k [a_ij + b_ij] = [k (a_ij + b_ij)] = [(k a_ij) + (k b_ij)]

= [k a_ij] + [k b_ij] = k [a_ij] + k [b_ij] = kA + kB

(ii) (k + l) A = (k + l) [a_ij]

= [(k + l) a_ij] = [k a_ij] + [l a_ij] = k [a_ij] + l [a_ij] = k A + l A.

mnkgj.k 8;fn

8 0 2 2

A 4 2 , B 4 2

3 6 5 1

−

   

   

=_ − _ =_{ }

  − 

   

rFkk 2A + 3X = 5B fn;k gks rks vkO;wg X

Kkr dhft,A

gy fn;k gS 2A + 3X = 5B

;k 2A + 3X – 2A = 5B – 2A

;k 2A – 2A + 3X = 5B – 2A (vkO;wg ;ksx Øe&fofues; gS)

;k O + 3X = 5B – 2A (– 2A, vkO;wg 2A dk ;ksx izfrykse gS)

;k 3X = 5B – 2A (O, ;ksx dk rRled gS)

;k X = 1

3 (5B – 2A)

;k

2 2 8 0

1

X 5 4 2 2 4 2

3

5 1 3 6

  −   

    

= _{  }− _ − _

 _{−   }

 

=

10 10 16 0

1

20 10 8 4

3

25 5 6 12

 −  − 

  _{+ −} 

   

_{−   − − }

=

10 16 10 0 1

20 8 10 4 3

25 6 5 12

− − +    _{− +}    − − −    = 6 10 1 12 14 3 31 7 − −       − −    = 10 2 3 14 4 3 31 7 3 3 −   −         _{− −}       

mnkgj.k 9X rFkk Y, Kkr dhft,] ;fn X Y 5 2 0 9

 

+ _{=  }

  rFkk

3 6 X Y 0 1   − _{=  } −

  gSA

gy ;gk¡ ij (X + Y) + (X – Y) = 5 2 3 6

0 9 0 1

  ₊ 

   ₋ 

   

;k (X + X) + (Y – Y) = 8 8

0 8       ⇒ 8 8 2X 0 8   =    

;k X = 1 8 8 4 4

0 8 0 4

2

   

=

   

   

lkFk gh (X + Y) – (X – Y) = 5 2 3 6

0 9 0 1

  ₋ 

   ₋ 

   

;k (X – X) + (Y + Y) = 5 3 2 6

0 9 1

− −    ₊    ⇒ 2 4 2Y 0 10 −   =    

;k Y = 1 2 4 1 2

0 10 0 5

2 − −     =        

mnkgj.k 10fuEufyf[kr lehdj.k ls xrFkk yosQ ekuksa dks Kkr dhft,%

5 3 4

2

7 3 1 2

x y −     +  ₋        = 7 6 15 14      

gy fn;k gS

5 3 4

2

7 3 1 2

x y −   ₊   ₋        = 7 6 15 14       ⇒

2 10 3 4 7 6

14 2 6 1 2 15 14

;k _{14 1}2x+₊3 _2y10 4_{− +}−_{6 2}

  =

7 6 15 14

 

 

  ⇒

2 3 6 7 6

15 2 4 15 14

x

y

+

   

=

 ₋   

   

;k 2x + 3 = 7 rFkk 2y – 4 = 14 (D;ksa?)

;k 2x = 7 – 3 rFkk 2y = 18

;k x = 4

2 rFkk y =

18 2

vFkkZr~ x = 2 rFkk y = 9

mnkgj.k 11nks fdlku jkefd'ku vkSj xqjpju flag osQoy rhu izdkj osQ pkoy tSls cklerh]

ijey rFkk umjk dh [ksrh djrs gSaA nksuksa fdlkuksa }kjk] flracj rFkk vDrwcj ekg esa] bl izdkj osQ pkoy dh fcØh (#i;ksa esa) dks] fuEufyf[kr A r Fkk B vkO;wgksa esa O;Dr fd;k x;k gS%

(i) izR;sd fdlku dh izR;sd izdkj osQ pkoy dh flracj rFkk vDrwcj dh lfEefyr fcØh

Kkr dhft,A

(ii) flracj dh vis{kk vDrwcj esa gqbZ fcØh esa deh Kkr dhft,A

(iii) ;fn nksuksa fdlkuksa dks oqQy fcØh ij 2% ykHk feyrk gS] rks vDrwcj esa izR;sd izdkj osQ

pkoy dh fcØh ij izR;sd fdlku dks feyus okyk ykHk Kkr dhft,A

gy

(i) izR;sd fdlku dh izR;sd izdkj osQ pkoy dh flracj rFkk vDrwcj esa izR;sd izdkj osQ

(ii) flracj dh vis{kk vDrwcj esa gqbZ fcØh esa deh uhps nh xbZ gS]

(iii) B dk 2% = 2 B

100× = 0. 02 × B

= 0.02

=

vr% vDrwcj ekg esa] jkefd'ku] izR;sd izdkj osQ pkoy dh fcØh ij Øe'k% Rs100]

Rs 200] rFkk Rs 120 ykHk izkIr djrk gS vkSj xqjpju flag] izR;sd izdkj osQ pkoy dh fcØh

ij Øe'k% Rs 400] Rs 200 rFkk Rs 200 ykHk vftZr djrk gSA

3.4.5 vkO;wgksa dk xq.ku (Multiplication of matrices)

eku yhft, fd ehjk vkSj unhe nks fe=k gSaA ehjk 2 dye rFkk 5 dgkuh dh iqLrosaQ [kjhnuk pkgrh gSa] tc fd unhe dks 8 dye rFkk 10 dgkuh dh iqLrdksa dh vko';drk gSA os nksuksa ,d nqdku ij (dher) Kkr djus osQ fy, tkrs gSa] tks fuEufyf[kr izdkj gS%

dye & izR;sd Rs 5] dgkuh dh iqLrd & izR;sd Rs 50 gSA

mu nksuksa esa ls izR;sd dks fdruh /ujkf'k [kpZ djuh iM+sxh\ Li"Vr;k] ehjk dks

Rs (5 × 2 + 50 × 5) vFkkZr~] Rs 260 dh vko';drk gS] tcfd unhe dks Rs (8 × 5 + 50 × 10)

vFkkZr~ Rs 540 dh vko;drk gSA ge mi;qZDr lwpuk dks vkO;wg fu:i.k esa fuEufyf[kr izdkj ls izdV

vko';drk izfr ux nke (#i;ksa esa) vko';d /ujkf'k (#i;ksa esas)

2 5 8 10

 

 

 

5 50      

5 2 5 50 260 8 5 10 50 540

× + ×

   

=

 _{× + ×}   

   

eku yhft, fd muosQ }kjk fdlh vU; nqdku ij Kkr djus ij Hkko fuEufyf[kr izdkj gSa% dye & izR;sd Rs 4] dgkuh dh iqLrd & izR;sd Rs 40

vc] ehjk rFkk unhe }kjk [kjhnkjh djus osQ fy, vko';d /ujkf'k Øe'k% Rs (4 × 2 + 40 × 5) = Rs 208 rFkk Rs (8 × 4 + 10 × 40) = Rs 432 gSA

iqu% mi;qZDr lwpuk dks fuEufyf[kr <ax ls fu:fir dj ldrs gSa%

vko';drk izfr ux nke (#i;ksa esa) vko';d /ujkf'k (#i;ksa esas)

2 5 8 10

 

 

 

4 40      

4 2 40 5 208 8 4 10 4 0 432

× + ×

  ₌ 

 _{× + ×}   

   

vc] mi;qZDr nksuksa n'kkvksa esa izkIr lwpukvksa dks ,d lkFk vkO;wg fu:i.k }kjk fuEufyf[kr izdkj ls izdV dj ldrs gSa%

vko';drk izfr ux nke (#i;ksa esa) vko';d /ujkf'k (#i;ksa esas)

2 5 8 10

 

 

 

5 4

50 40

 

 

 

5 2 5 50 4 2 40 5 8 5 10 5 0 8 4 10 4 0

× + × × + ×

 

 _{× + × × + ×} 

 

= 260 208 540 432

 

 

 

mi;qZDr fooj.k vkO;wgksa osQ xq.ku dk ,d mnkgj.k gSA ge ns[krs gSa fd vkO;wgksa A rFkk B osQ

xq.ku osQ fy,] A esa LraHkksa dh la[;k B esa iafDr;ksa dh la[;k osQ cjkcj gksuh pkfg,A blosQ vfrfjDr

xq.kuiQy vkO;wg (Product matrix) osQ vo;oksa dks izkIr djus osQ fy,] ge A dh iafDr;ksa rFkk B osQ LraHkksa dks ysdj] vo;oksa osQ Øekuqlkj (Element–wise) xq.ku djrs gSa vkSj rnksijkar bu

xq.kuiQyksa dk ;ksxiQy Kkr djrs gSaA vkSipkfjd :i ls] ge vkO;wgksa osQ xq.ku dks fuEufyf[kr rjg ls ifjHkkf"kr djrs gSa%

nks vkO;wgksa A rFkk B dk xq.kuiQy ifjHkkf"kr gksrk gS] ;fn A esa LraHkksa dh la[;k] B esa iafDr;ksa

dh la[;k osQ leku gksrh gSA eku yhft, fd A = [a_ij] ,d m × ndksfV dk vkO;wg gS vkSj

B = [b_jk] ,dn × pdksfV dk vkO;wg gSA rc vkO;wgksa A rFkk B dk xq.kuiQy ,d m × p dksfV

dk vkO;wg C gksrk gSA vkO;wg C dk (i, k)ok¡ vo;o c_ikizkIr djus osQ fy, ge A dh i oha iafDr

vkSj B osQ kosa LraHk dks ysrs gS vkSj fiQj muosQ vo;oksa dk Øekuqlkj xq.ku djrs gSaA rnksijkUr bu

A = [a_ij]_{m × n}, B = [b_jk]_{n × p}gS rks Adh i oha iafDr [a_i1a_i2 ... a_in] rFkk B dk kok¡ LraHk

1 2 . . . k k

nk

b b

b

 

 

 

 

 

 

 

gSa, rc c_ik = a_i1b_1k + a_i2 b_2k + a_i3 b_3k + ... + a_inb_nk = 1 n

ij jk j

a b =

∑

vkO;wg C = [c_ik]_{m × p}, A rFkk B dk xq.kuiQy gSA

mnkgj.k osQ fy,] ;fn C 1 1 2

0 3 4 −

 

=  

  rFkk

2 7 1

D 1

5 4

 

 

= −_{ }

 − 

 

gS rks

xq.kuiQy

2 7

1 1 2

CD CD 1 1

0 3 4

5 4

 

−

 _{  }

=_{  }− _

 _{  − }

 

ifjHkkf"kr gSrFkk ,d 2 × 2 vkO;wg gS ftldh

izR;sd izfof"V C dh fdlh iafDr dh izfof"V;ksa dh D osQ fdlh LraHk dh laxr izfof"V;ksa osQ

xq.kuiQyksa osQ ;ksxiQy osQ cjkcj gksrh gSA bl mnkgj.k esa ;g pkjksa ifjdyu fuEufyf[kr gSa]

vr% CD 13 2

17 13 −

 

=  ₋ 

mngkj.k 12 ;fn A 6 9 B 2 6 0

2 3 7 9 8

   

=_{ } =_{ }

  rFkk   gS rks AB Kkr dhft,A

gyvkO;wg A esa 2 LraHk gSa tks vkO;wg B dh iafDr;ksa osQ leku gSaA vr,o AB ifjHkkf"kr gSA vc

AB = 6(2) 9(7) 6(6) 9(9) 6(0) 9(8) 2(2) 3(7) 2(6) 3(9) 2(0) 3(8)

+ + +

 

 _{+ + +} 

 

= 12 63 36 81 0 72 4 21 12 27 0 24

+ + +

 

 _{+ + +} 

  =

75 117 72 25 39 24

 

 

 

A

fVIi.kh ;fn AB ifjHkkf"kr gS rks ;g vko';d ugha gS fd BA Hkh ifjHkkf"kr gksA mi;qZDr

mnkgj.k esa AB ifjHkkf"kr gS ijarq BA ifjHkkf"kr ugha gS D;ksafd B esa 3 LraHk gSa tcfd A esa osQoy

2 iafDr;k¡ (3 iafDr;k¡ ugha) gSaA ;fn A rFkk B Øe'k% m × nrFkk k × ldksfV;ksa osQ vkO;wg

gSa rks AB rFkk BA nksuksa gh ifjHkkf"kr gSa ;fn vkSj osQoy ;fn n = k rFkk l = m gksA fo'ks"k

:i ls] ;fn A vkSj B nksuksa gh leku dksfV osQ oxZ vkO;wg gSa] rks AB rFkk BA nksuksa ifjHkkf"kr

gksrs gSaA

vkO;wgksa osQ xq.ku dh vØe&fofues;rk (Non-Commutativity of multiplication of matrices)

vc ge ,d mnkgj.k osQ }kjk ns[ksaxs fd] ;fn AB rFkk BA ifjHkkf"kr Hkh gksa] rks ;g vko';d

ugha gS fd AB = BA gksA

mnkgj.k 13 ;fn

2 3

1 2 3

A B 4 5

4 2 5

2 1

 

−

  _{ }

=_{ } =_{ }

−

  _{ }

 

vkSj , rks AB rFkk BA Kkr dhft,A n'kkZb, fd

AB ≠ BA

gy D;ksafd fd A ,d 2 × 3 vkO;wg gS vkSj B ,d 3 × 2 vkO;wg gS] blfy, AB rFkk BA nksuksa

gh ifjHkkf"kr gSa rFkk Øe'k% 2 × 2 rFkk 3 × 3, dksfV;ksa osQ vkO;wg gSaA uksV dhft, fd

AB =

2 3 1 2 3

4 5 4 2 5

2 1

 

−

 _{  }

_{−   }

 _{  }

 

= 2 8 6 3 10 3 0 4

8 8 10 12 10 5 10 3

− + − + −

  ₌ 

_{− + + − + +}   

   

vkSj BA =

2 3 2 12 4 6 6 15

1 2 3

4 5 4 20 8 10 12 25

4 2 5

2 1 2 4 4 2 6 5

− − + +

   

−

 

  ₌ _{− − + +} 

 

 _{− }  

   − − + + 

   

10 2 21

16 2 37 2 2 11 −

 

 

= −_{ }

 − − 

 

mi;qZDr mnkgj.k esa AB rFkk BA fHkUu&fHkUu dksfV;ksa osQ vkO;wg gSa vkSj blfy, AB ≠ BA

gSAijarq dksbZ ,slk lksp ldrk gS fd ;fn AB rFkk BA nksuksa leku dksfV osQ gksrs rks laHkor% os

leku gkasxsA ¯drq ,slk Hkh ugha gSA ;gk¡ ge ,d mnkgj.k ;g fn[kykus osQ fy, ns jgs gSa fd ;fn

AB rFkk BA leku dksfV osQ gksa rks Hkh ;g vko';d ugha gS fd os leku gksaA

mnkgj.k 14;fn A = 1 0

0 1

 

 ₋ 

  rFkk

0 1 B

1 0

 

=  

  gS rks

0 1 AB

1 0

 

= ₋ 

 

vkSj BA = 0 1

1 0

−

 

 

  gSA Li"Vr;k AB ≠ BA gSA

vr% vkO;wg xq.ku Øe&fofues; ugha gksrk gSA

A

fVIi.kh bldk rkRi;Z ;g ugha gS fd A rFkk B vkO;wgksa osQ mu lHkh ;qXeksa osQ fy,] ftuosQ

fy, AB rFkk BA ifjHkkf"kr gS] AB ≠ BA gksxkA mnkgj.k osQ fy,

;fn A 1 0 , B 3 0

0 2 0 4

   

=_{ } =_{ }

   , rks AB = BA = 3 0 0 8

 

 

 

è;ku nhft, fd leku dksfV osQ fod.kZ vkO;wgksa dk xq.ku Øe&fofues; gksrk gSA

nks 'kwU;srj vkO;wgksa osQ xq.kuiQy osQ :i esa 'kwU; vkO;wg%(Zero matrix as the product

of two non-zero matrices)

gesa Kkr gS fd nks okLrfod la[;kvksa arFkkbosQ fy,] ;fn ab = 0 gS rks ;k rks a = 0 vFkok

b = 0 gksrk gSA ¯drq vkO;wgksa osQ fy, ;g vfuok;Zr% lR; ugha gksrk gSA bl ckr dks ge ,d mnkgj.k

}kjk ns[ksaxsA

mnkgj.k 15 ;fn A 0 1

0 2

−

 

=  

  rFkk

3 5 B

0 0

 

=  

  gS rks AB dk eku Kkr dhft,

gy ;gk¡ ij AB 0 1 3 5 0 0

0 2 0 0 0 0

−

     

=_{    }= _

     

vr% ;fn nks vkO;wgksa dk xq.kuiQy ,d 'kwU; vkO;wg gS rks vko';d ugha gS fd muesa ls ,d vkO;wg vfuok;Zr% 'kwU; vkO;wg gksA

3.4.6 vkO;wgksa osQ xq.ku osQ xq.k/eZ (Properties of multiplication of matrices)

vkO;wgksa osQ xq.ku osQ xq.k/eks± dk ge uhps fcuk mudh miifÙk fn, mYys[k dj jgs gSa%

1. lkgp;Z fu;e%fdUgha Hkh rhu vkO;wgksa A, B rFkk C osQ fy,

2. forj.k fu;e% fdUgha Hkh rhu vkO;wgksa A, B rFkk C osQ fy,

(i) A (B+C) = AB + AC

(ii) (A+B) C = AC + BC, tc Hkh lehdj.k osQ nksuksa i{k ifjHkkf"kr gksrs gSaA

3. xq.ku osQ rRled dk vfLrRo % izR;sd oxZ vkO;wg A osQ fy, leku dksfV osQ ,d vkO;wg

I dk vfLrRo bl izdkj gksrk gS] fd IA = AI = A

vc ge mnkgj.kksa osQ }kjk mi;qZDr xq.k/ek± dk lR;kiu djsaxsA

mnkgj.k 16 ;fn

1 1 1 1 3

1 2 3 4

A 2 0 3 , B 0 2 C

2 0 2 1

3 1 2 1 4

−

   

−

 

   

=_{ } =_{  =  }

−

 

 −  − 

   

rFkk rks A(BC)

rFkk (AB)C Kkr dhft, vkSj fn[kykb, fd (AB)C = A(BC) gSA

gy ;gk¡ AB =

1 1 1 1 3 1 0 1 3 2 4 2 1

2 0 3 0 2 2 0 3 6 0 12 1 18

3 1 2 1 4 3 0 2 9 2 8 1 15

− + + + −

       

    _{= + − + +}  _{= −} 

       

 −  −   + − − +   

       

(AB) (C) =

2 2 4 0 6 2 8 1

2 1

1 2 3 4

1 18 1 36 2 0 3 36 4 18

2 0 2 1

1 15 1 30 2 0 3 30 4 15

+ + − − +

 

  ₋

 _{  }

₋  _{= − + − + − − +}

 _{  }

 _{ −}

  

  + + − − +

   

=

4 4 4 7

35 2 39 22

31 2 27 11 −

 

 _{− −} 

 

 − 

 

vc BC =

1 6 2 0 3 6 4 3

1 3

1 2 3 4

0 2 0 4 0 0 0 4 0 2

2 0 2 1

1 4 1 8 2 0 3 8 4 4

+ + − − +

 

 

−

 _{  }

  _{= + + − +}

 _{  }

 _{ −}

  

−  − + − + − − +

   

=

7 2 3 1

4 0 4 2

7 2 11 8

− −

 

 ₋ 

 

 − − 

vr,o A(BC) =

7 2 3 1

1 1 1

2 0 3 4 0 4 2

3 1 2 7 2 11 8

− −

−  

 

 

  ₋

 

 

 

 −  − −

   

=

7 4 7 2 0 2 3 4 11 1 2 8

14 0 21 4 0 6 6 0 33 2 0 24

21 4 14 6 0 4 9 4 22 3 2 16

+ − + + − − + − + −

 

 _{+ + + − − + − − + +} 

 

 − + + − − + − − − + 

 

=

4 4 4 7

35 2 39 22

31 2 27 11

−

 

 _{− −} 

 

 − 

 

Li"Vr;k] (AB) C = A (BC)

mnkgj.k 17 ;fn

0 6 7 0 1 1 2

A 6 0 8 , B 1 0 2 , C 2

7 8 0 1 2 0 3

     

     

= −_{ } =_{ } = −_{ }

 −     

     

rks AC, BC rFkk (A + B)C dk ifjdyu dhft,A ;g Hkh lR;kfir dhft, fd (A + B) C = AC + BC

gy

0 7 8

A+ B 5 0 10 8 6 0

 

 

= −_{ }

 − 

 

vr,o] (A + B) C =

0 7 8 2 0 14 24 10

5 0 10 2 10 0 30 20

8 6 0 3 16 12 0 28

− +

       

₋    _{− = − + +}   ₌

       

 −     + +   

       

blosQ vfrfjDr AC =

0 6 7 2 0 12 21 9

6 0 8 2 12 0 24 12

7 8 0 3 14 16 0 30

− +

       

₋    _{− = − + +}   ₌

       

 −     + +   

vkSj BC =

0 1 1 2 0 2 3 1

1 0 2 2 2 0 6 8

1 2 0 3 2 4 0 2

− +

       

  ₋  _{= + +}   ₌

       

     − +   −

       

blfy, AC + BC =

9 1 10

12 8 20

30 2 28

           _{+ =}            −      

Li"Vr;k (A + B) C = AC + BC

mnkgj.k 18;fn

1 2 3

A 3 2 1

4 2 1

 

 

=_ − _

 

 

gS rks n'kkZb, fd A3 _{– 23A – 40I = O}

gy ge tkurs gSa fd 2

1 2 3 1 2 3 19 4 8

A A.A 3 2 1 3 2 1 1 12 8

4 2 1 4 2 1 14 6 15

     

     

= =_ − _{ } − _{ }= _

     

     

blfy, A3 _{= A A}2 ₌

1 2 3 19 4 8 63 46 69

3 2 1 1 12 8 69 6 23

4 2 1 14 6 15 92 46 63

     

 ₋    _{= −} 

     

     

     

vc A3 _{– 23A – 40I =}

63 46 69 1 2 3 1 0 0

69 6 23 – 23 3 2 1 – 40 0 1 0

92 46 63 4 2 1 0 0 1

     

 ₋   ₋   

     

     

     

=

63 46 69 23 46 69 40 0 0

69 6 23 69 46 23 0 40 0

92 46 63 92 46 23 0 0 40

− − − −

     

 ₋  _{+ − −}  _{+ −} 

     

  − − −   − 

     

=

63 23 40 46 46 0 69 69 0 69 69 0 6 46 40 23 23 0

92 92 0 46 46 0 63 23 40

− − − + − +

 

 _{− + − + − − +} 

 

 − + − + − − 

 

=

0 0 0

0 0 0 O

0 0 0

 

 _{ =}

 

 

mnkgj.k 19 fdlh fo/ku lHkk pquko osQ nkSjku ,d jktuSfrd ny us vius mEehnokj osQ izpkj

gsrq ,d tu laioZQ iQeZ dks BsosQ ij vuqcaf¼r fd;kA izpkj gsrq rhu fof/;ksa }kjk laioZQ LFkkfir djuk fuf'pr gqvkA ;s gSa% VsyhiQksu }kjk] ?kj&?kj tkdj rFkk ipkZ forj.k }kjkA izR;sd laioZQ dk 'kqYd (iSlksa esa) uhps vkO;wg A esa O;Dr gS]

A = 40

100

50

 

 

 

 

 

ifzr liaoQZ ewY;

Vys hiQkus }kjk ?kj tkdj

ipkZ }kjk

X rFkk Y nks 'kgjksa esa] izR;sd izdkj osQ lEidks± dh la[;k vkO;wg

1000 500 5000 X B

Y 3000 1000 10, 000

→

 

=  _{ →}

 

Vys hiQksu ?kj tkdj ipkZ }kjk

esa O;Dr gSA X rFkk Y 'kgjksa esa jktuSfrd ny }kjk O;; dh

xbZ oqQy /ujkf'k Kkr dhft,A

gy ;gk¡ ij

BA = 40, 000 50,000 250, 000 X Y

120,000 +100,000 +500,000

+ + →

 

 _{ →}

 

= 340, 000 X Y 720,000

→

 

 _{ →}

 

vr% ny }kjk nksuksa 'kgjksa esa O;; dh xbZ oqQy /ujkf'k Øe'k% 3]40]000 iSls o 7]20]000 iSls vFkkZr~ Rs 3400 rFkk Rs 7200 gSaA

iz'ukoyh 3-2

1. eku yhft, fd A 2 4 , B 1 3 , C 2 5

3 2 2 5 3 4

−

     

=_{ } =_{ } =_{ }

−

     , rksfuEufyf[kr Kkr dhft,%

(i) A + B (ii) A – B (iii) 3A – C

2. fuEufyf[kr dks ifjdfyr dhft,%

(i) a b a b

b a b a

   

+

₋   

    (ii)

2 2 2 2

2 2 2 2

2 2

2 2

a b b c ab bc

ac ab

a c a b

 + +    +  _{  } − − + +       (iii)

1 4 6 12 7 6

8 5 16 8 0 5

2 8 5 3 2 4

− −       ₊              (iv)

2 2 2 2

2 2 2 2

cos sin sin cos

sin cos cos sin

x x x x

x x x x

   

+

   

   

   

3. funf'kZr xq.kuiQy ifjdfyr dhft,%

(i) a b a b

b a b a

−

   

₋   

    (ii)

[

]

1

2 2 3 4

3          

(iii) 1 2 1 2 3

2 3 2 3 1

−

   

   

   

(iv)

2 3 4 1 3 5

3 4 5 0 2 4

4 5 6 3 0 5

−                     (v) 2 1

1 0 1 3 2

1 2 1 1 1          _{ − } −    (vi) 2 3 3 1 3

1 0 1 0 2

3 1 −   −  _{  } _{−   }  _{  }   . 4. ;fn

1 2 3 3 1 2 4 1 2

A 5 0 2 , B 4 2 5 C 0 3 2

1 1 1 2 0 3 1 2 3

− −             =_{ } =_{ } =_{ }  −     −       

rFkk , rks (A+B) rFkk

(B – C) ifjdfyr dhft,A lkFk gh lR;kfir dhft, fd A + (B – C) = (A + B) – C.

5. ;fn

2 5 2 3

1 1

3 3 5 5

1 2 4 1 2 4

A B

3 3 3 5 5 5

7 2 7 6 2

2

3 3 5 5 5

                =_{ } =_{ }                

6. ljy dhft,] cos cos sin + sin sin cos sin cos cos sin

θ θ θ − θ

   

θ_{ } θ_{ }

− θ θ θ θ

   

7. X rFkk Y Kkr dhft, ;fn

(i) X + Y 7 0 X – Y 3 0

2 5 0 3

   

=_{ } =_{ }

  rFkk  

(ii) 2X + 3Y 2 3 3X 2Y 2 2

4 0 1 5

−

   

=_{ } + =_{ }

−

  rFkk  

8. X rFkk Y Kkr dhft, ;fn Y = 3 2

1 4

 

 

  rFkk 2X + Y = 1 0

3 2

 

₋ 

 

9. x

rFkk

yKkr dhft, ;fn 2 1 3 0 5 6

0 1 2 1 8

y x

     

+ =

     

     

10. iznÙk lehdj.k dks x, y, z _rFkk tosQ fy, gy dhft, ;fn

1 1 3 5

2 3 3

0 2 4 6

x z y t

−

  ₊  ₌  

     

     

11. ;fn 2 1 10

3 1 5

x _{ }+y   _{   }− =

      gS rks xrFkk yosQ eku Kkr dhft,A

12. ;fn 3 6 4

1 2 3

x y x x y

z w w z w

+

  ₌  ₊ 

  ₋   ₊ 

      gS rks x, y, zrFkk wosQ ekuksa dks Kkr

dhft,A

13. ;fn

cos sin 0

F ( ) sin cos 0

0 0 1

x x

x x x

−

 

 

=  

 

 

gS rks fl¼ dhft, fd F(x) F(y) = F(x + y)

14. n'kkZb, fd

(i) 5 1 2 1 2 1 5 1

6 7 3 4 3 4 6 7

− −

    _≠   

       

(ii)

1 2 3 1 1 0 1 1 0 1 2 3

0 1 0 0 1 1 0 1 1 0 1 0

1 1 0 2 3 4 2 3 4 1 1 0

− −

       

   ₋  _{≠ −}   

       

       

       

15. ;fn

2 0 1

A 2 1 3

1 1 0

 

 

=  

 − 

 

gS rks A2 _{– 5A + 6I,} dk eku Kkr dhft,A

16. ;fn

1 0 2

A 0 2 1

2 0 3

 

 

=  

 

 

gS rks fl¼ dhft, fd A3 _{– 6A}2 _{+ 7A + 2I = 0}

17. ;fn A 3 2 I = 1 0

4 2 0 1

−

   

=  ₋   

  rFkk   ,oa A

2 _{= kA – 2I} gks rks _kKkr dhft,A

18. ;fn

0 tan

2 A

tan 0

2

α

 ₋ 

 

=  

α

 

 

 

rFkk I dksfV 2 dk ,d rRled vkO;wg gSA rks fl¼ dhft,

fd I + A = (I – A) cos sin sin cos

α − α

 

 _{α α}

 

19. fdlh O;kikj la?k osQ ikl 30]000 #i;ksa dk dks"k gS ftls nks fHkUu&fHkUu izdkj osQ ckaMksa

esa fuosf'kr djuk gSA izFke ckaM ij 5% okf"kZd rFkk f}rh; ckaM ij 7% okf"kZd C;kt izkIr

gksrk gSA vkO;wg xq.ku osQ iz;ksx }kjk ;g fu/kZfjr dhft, fd 30]000 #i;ksa osQ dks"k dks nks izdkj osQ ckaMksa esa fuos'k djus osQ fy, fdl izdkj ck¡Vas ftlls O;kikj la?k dks izkIr oqQy okf"kZd C;kt

(a) Rs 1800 gksA (b) Rs 2000 gksA

20. fdlh LowQy dh iqLrdksa dh nqdku esa 10 ntZu jlk;u foKku] 8 ntZu HkkSfrd foKku rFkk

10 ntZu vFkZ'kkL=k dh iqLrosaQ gSaA bu iqLrdksa dk foØ; ewY; Øe'k% Rs 80] Rs 60 rFkk Rs 40 izfr iqLrd gSA vkO;wg chtxf.kr osQ iz;ksx }kjk Kkr dhft, fd lHkh iqLrdksa dks

cspus ls nqdku dks oqQy fdruh /ujkf'k izkIr gksxhA

eku yhft, fd X, Y, Z, W rFkk P Øe'k% 2 × n, 3 × k, 2 × p, n × 3 rFkk p × k, dksfV;ksa

21. PY + WY osQ ifjHkkf"kr gksus osQ fy, n, krFkk p ij D;k izfrca/ gksxk\

(A) k = 3, p = n (B) kLosPN gS , p = 2

(C) pLosPN gS, k = 3 (D) k = 2, p = 3

22. ;fn n = p, rks vkO;wg 7X – 5Z dh dksfV gSA

(A) p × 2 (B) 2 × n (C) n × 3 (D) p × n

3.5.

vkO;wg dk ifjorZ

(Transpose of a Matrix)

bl vuqPNsn esa ge fdlh vkO;wg osQ ifjorZ rFkk oqQN fo'ks"k izdkj osQ vkO;wgksa] tSls lefer

vkO;wg (Symmetric Matrix) rFkk fo"ke lefer vkO;wg (Skew Symmetric Matrix) osQ

ckjs esa tkusaxsA

ifjHkk"kk 3 ;fn A = [a_ij] ,d m × ndksfV dk vkO;wg gS rks A dh iafDr;ksa rFkk LraHkksa dk ijLij

fofue; (Interchange) djus ls izkIr gksus okyk vkO;wg A dk ifjorZ (Transpose) dgykrk

gSA vkO;wg A osQ ifjorZ dks A′(;k AT₎ ls fu:fir djrs gSaA nwljs 'kCnksa esa] ;fn

A = [a_ij]_{m × n}, rks A′ = [a_ji]_{n × m}gksxkA mnkgj.kkFkZ] ;fn

A =

2 3 3 2

3 _{5 3 3 0}

3 1 A ₁

5 1

0 1 ₅

5

× ×

  _{ }

  _{′ = }

−

  _{ }

 ₋  _{ }

 

 

gks rks _gksxkA

vkO;wgksa osQ ifjorZ osQ xq.k/eZ (Properties of transpose of matrices)

vc ge fdlh vkO;wg osQ ifjorZ vkO;wg osQ fuEufyf[kr xq.k/eks± dks fcuk miifÙk fn, O;Dr djrs gSaA budk lR;kiu mi;qDr mnkgj.kksa }kjk fd;k tk ldrk gSaA mi;qDr dksfV osQ fdUgha vkO;wgksa

A rFkk B osQ fy,

(i) (A′)′ = A (ii) (kA)′ = kA′ (tgk¡kdksbZ vpj gSA) (iii) (A + B)′ = A′ + B′ (iv) (A B)′ = B′ A′

mnkgj.k 20 ;fn A 3 3 2 B 2 1 2

1 2 4

4 2 0

   − 

=_{ } =_{ }

 

  rFkk rks fuEufyf[kr dks lR;kfir

dhft,%

gy

(i) ;gk¡

A = ( )

3 4

3 3 2 3 3 2

A 3 2 A A

4 2 0 4 2 0

2 0

 

  _′   _{′ ′}  

⇒ = ⇒ = =

     

  _ _  

vr% (A′)′ = A (ii) ;gk¡

A = 3 3 2 ,

4 2 0

 

 

  B =

2 1 2 5 3 1 4

A B

1 2 4 _{5 4 4}

 

−

 _⇒ −

+ =  

 

   

vr,o (A + B)′ =

5 ₅

3 1 4

4 4

 

 ₋ 

 

 

 

vc A′ =

3 4 2 1

3 2 , B 1 2

2 0 2 4

  _{ }

  _{′ = − }

  _{ }

  _{ }

   

vr,o A′ + B′ =

5 ₅

3 1 4

4 4

 

 ₋ 

 

 

 

vr% (A + B)′ = A′ + B′ (iii) ;gk¡

kB = k 2 1 2 2 2

1 2 4 2 4

k k k

k k k

− −

  ₌ 

   

   

rc (kB)′ =

2 2 1

2 1 2 B

2 4 2 4

k k

k k k k

k k

   

₋ _{= −} _{= ′}

   

   

   

mnkgj.k 21 ;fn

[

]

2

A 4 , B 1 3 6

5 −

 

 

=_{ } = −

 

 

gS rks lR;kfir dhft, (AB)′ = B′A′ gSA

gy ;gk¡

A =

[

]

2

4 , B 1 3 6

5 −

 

  _{= −}

     

blfy, AB =

[

]

2

4 1 3 6

5 −

 

  ₋

     

=

2 6 12

4 12 24

5 15 30

− −

 

 ₋ 

 

 − 

 

vr% (AB)′ =

2 4 5

6 12 15

12 24 30

−

 

₋ 

 

 − − 

 

vc A′ = [–2 4 5] ,

1

B 3

6

 

 

= ′  

− 

 

blfy, B′A′ =

[

]

1 2 4 5

3 2 4 5 6 12 15 (AB)

6 12 24 30

−

   

  _{− = −} _{= ′}

   

−   − − 

   

Li"Vr;k (AB)′ = B′A′

3.6

lefer rFkk fo"ke lefer vkO;wg

(Symmetric and Skew Symmetric Matrices)

ifjHkk"kk 4,d oxZ vkO;wg A = [a_ij]lefer dgykrk gS ;fn A′ = A vFkkZr~ iojosQ gj laHko

ekuksa osQ fy, [a_ij] = [a_ji] gksA

mnkgj.k osQ fy,]

3 2 3

A 2 1.5 1

3 1 1

 

 

=_ − − _

 ₋ 

 

ifjHkk"kk 5 ,d oxZ vkO;wg A = [a_ij]fo"ke lefer vkO;wg dgykrk gS] ;fn A′ = – A, vFkkZr~

i rFkk j osQ gj laHko ekuksa osQ fy, a_ji = – a_ij gksA vc] ;fn ge i = jj[ksa] rks a_ii = – a_iigksxkA

vr% 2a_ii = 0 ;ka_ii = 0 leLr iosQ fy,A

bldk vFkZ ;g gqvk fd fdlh fo"ke lefer vkO;wg osQ fod.kZ osQ lHkh vo;o 'kwU; gksrs

gSaA mnkgj.kkFkZ vkO;wg 0

B 0

0

e f

e g

f g

 

 

= −_{ }

− − 

 

,d fo"ke lefer vkO;wg gS] D;ksafd B′ = – B gSA

vc] ge lefer rFkk fo"ke lefer vkO;wgksa osQ oqQN xq.k/eks± dks fl¼ djsaxsA

izes; 1 okLrfod vo;oksa okys fdlh oxZ vkO;wg A osQ fy, A + A′,d lefer vkO;wg rFkk

A – A′,d fo"ke lefer vkO;wg gksrs gSaA

miifÙkeku yhft, fd B = A + A′rc

B′ = (A + A′)′

= A′ + (A′)′ (D;ksafd (A + B)′ = (A′ + B′) = A′ + A(D;ksafd (A′)′ = A)

= A + A′ (D;ksafd A + B = B + A) = B

blfy, B = A + A′,d lefer vkO;wg gSA

vc eku yhft, fd C = A – A′

C′ = (A – A′)′ = A′ – (A′)′ (D;ksa?) = A′ – A (D;ksa?)

= – (A – A′) = – C

vr% C = A – A′,d fo"ke lefer vkO;wg gSA

izes; 2 fdlh oxZ vkO;wg dks ,d lefer rFkk ,d fo"ke lefer vkO;wgksa osQ ;ksxiQy osQ :i

esa O;Dr fd;k tk ldrk gSA

miifÙkeku yhft, fd A ,d oxZ vkO;wg gSA ge fy[k ldrs gSa fd

1 1

A (A A ) (A A )

2 ′ 2 ′

izes; 1 }kjk gesa Kkr gS fd (A + A′) ,d lefer vkO;wg rFkk (A – A′) ,d fo"ke lefer

vkO;wg gSA D;ksafd fdlh Hkh vkO;wg A osQ fy, (kA)′ = kA′gksrk gSA blls fu"d"kZ fudyrk gS fd

1

(A A )

2 + ′ lefer vkO;wg rFkk

1

(A A )

2 − ′ fo"ke lefer vkO;wg gSA vr% fdlh oxZ vkO;wg dks

,d lefer rFkk ,d fo"ke lefer vkO;wgksa osQ ;ksxiQy osQ :i esa O;Dr fd;k tk ldrk gSA

mnkgj.k 22 vkO;wg

2 2 4

B 1 3 4

1 2 3

− −     = −_{ }  − −   

dks ,d lefer vkO;wg rFkk ,d fo"ke lefer

vkO;wg osQ ;ksxiQy osQ :i esa O;Dr dhft,A

gy ;gk¡ B′ =

2 1 1

2 3 2

4 4 3

−   _{− −}    − −   

eku yhft, fd P =

4 3 3

1 1

(B + B ) 3 6 2

2 2

3 2 6

− −     ′ = _− _ − −    = 3 3 2 2 2 3 3 1 2 3 1 3 2 − −       −     ₋   ₋      gSA

vc P′ =

3 3 2 2 2 3 3 1 2 3 1 3 2 − −       −     ₋   ₋      = P

vr% P = 1(B + B )

2 ′ ,d lefer vkO;wg gSA

lkFk gh eku yhft, Q =

1 5

0

2 2

0 1 5

1 1 1

(B – B ) 1 0 6 0 3

2 2 2

5 6 0

rc Q′ =

1 5 0

2 3 1

0 3 Q 2

5

3 0 2

 

 

 

−

 _{− = −}

 

₋ 

 

 

 

vr% Q = 1(B – B )

2 ′ ,d fo"ke lefer vkO;wg gSA

vc

3 3 1 5

2 0

2 2 2 2 _{2 2 4}

3 1

P + Q 3 1 0 3 1 3 4 B

2 2

1 2 3

3 5

1 3 3 0

2 2

− − − −

   

   

− −

 

   

− _{ }

   

=_{  }+ _= −_{ }=

 − − 

₋   _{  }

 ₋   ₋ 

   

   

vr% vkO;wg B ,d lefer vkO;wg rFkk ,d fo"ke lefer vkO;wg osQ ;ksxiQy osQ :i esa

O;Dr fd;k x;kA

iz'ukoyh 3-3

1. fuEufyf[kr vkO;wgksa esa ls izR;sd dk ifjorZ Kkr dhft,%

(i) 5 1 2 1          ₋  

(ii) 1 1

2 3

−

 

 

  (iii)

1 5 6

3 5 6

2 3 1

−

 

 

 

 ₋ 

 

2. ;fn

1 2 3 4 1 5

A 5 7 9 B 1 2 0

2 1 1 1 3 1

− − −

   

   

=_{ }